arXiv:gr-qc/0011030 v1 8 Nov 2000

The dipolar zero-modes of Einstein action:

An informal summary with some new issues

Giovanni Modanese

1

California Institute for Physics and Astrophysics

366 Cambridge Ave., Palo Alto, CA 94306

and

University of Bolzano – Industrial Engineering

Via Sernesi 1, 39100 Bolzano, Italy

To appear in the proceedings of “Gravitation and Cosmology: from the Hubble Radius to the

Planck Scale. A Symposium in Celebration of the 80th Birthday of Jean-Pierre Vigier”, World

Scientific Editor.

Abstract

We recall the main features of metric vacuum fluctuations which have the global property

R

d

4

x

p

g(x)R(x) = 0, even though R(x)

6= 0 locally. We stress that these fluctuations could

mediate an anomalous coupling between the gravitational field and coherent matter. Some
new issues are discussed: (1) these fluctuations still imply that

hT

µν

(x)

i = 0; (2) they are

not extrema of the action; (3) for finite duration, their volume in phase space is not zero; (4)
vacuum fluctuations of this kind are not allowed in QED; (5) their null-action property is a
nonperturbative feature; (6) any real pure e.m. field generates zero-modes of this kind, too,
up to terms of order G

2

.

In this note we describe a set of gravitational field configurations, called “dipolar zero

modes”, which have not been considered earlier in the literature. They give an exactly null
contribution to the pure Einstein action and can thus represent large vacuum fluctuations in the
quantized theory of gravity.

The basic idea behind dipolar fluctuations was discussed for the first time in our earlier

work on stability of Euclidean quantum gravity [1]; the Lorentzian case was treated in [2]. This
year we made the first explicit computations and we were able to set some lower bounds on the
strength of the fluctuations [3, 4]. Also we gave for the first time in Ref.s [3, 4]: (1) an estimate
of possible suppression effects by cosmological or R

2

-terms; (2) a computation of the total ADM

energy of the zero modes; (3) a clarification (in the Lorentzian case) of the influence of matter
fields on the fluctuations, with possible anomalous coupling.

Here, after a few general remarks about vacuum fluctuations and “spacetime foam” in

quantum gravity, we shall set out the general features of the dipolar fluctuations (Section 1) and
give some explicit order of magnitude calculations (Section 2). Then we shall show that a Λ-term
cuts, to some extent, the dipolar fluctuations (Section 3); this can lead in certain cases to an
anomalous coupling to matter (Section 4). In conclusion, we shall discuss a number of new topics
not addressed in Ref.s [3, 4].

1

e-mail address: [e-mail]

1

1

General features of the dipolar fluctuations

The functional integral of pure Einstein quantum gravity can be written as z =

R

d[g

µν

] exp(iS/¯

h),

with S =

R

d

4

x

p

g(x)R(x).

The “spacetime foam” [5] consists of fluctuations whose action does not exceed a quantity

of order ¯

h. This implies, for curvature fluctuations on a scale d, that

|R| < G/d

4

(according to

naive power counting) or

|R| < 1/(L

P

d) (according to numerical lattice estimates [6]). Therefore,

large fluctuations are expected to take place only at very small distances.

Since, however, the Einstein action is not positive definite, one can also expect some fluctua-

tions due to peculiar cancellations of distinct contributions to the action, which are by themselves
larger than ¯

h.

In order to work this out explicitely, let us consider the Einstein equations with an auxiliary

source T

µν

:

R

µν

(x)

−

1
2

g

µν

(x)R(x) =

−8πGT

µν

(x)

(1)

and their trace

R(x) = 8πGg

µν

(x)T

µν

(x)

≡ 8πGTrT (x)

(2)

Then consider a solution g

µν

(x) of (1) with any source satisfying the condition

Z

d

4

x

q

g(x)TrT (x) = 0

(3)

In view of (2), this metric has zero action. We have constructed in this way a zero mode of the
pure Einstein action. The source is unphysical, but it is “forgotten” after obtaining the metric.
Condition (3) means in fact that it is a “dipolar” source, with a compensation between regions
having positive and negative mass-energy density. Since this auxiliary source is used to construct
a virtual field configuration, we shall sometimes call it a “virtual source”.

2

Explicit computation to order G

2

We have then found the following “recipe” for constructing dipolar zero-modes of the pure Einstein
action: given any source with zero integral (condition (3)), one solves the Einstein equations and
finds the corresponding metric.

Note, however, that the condition on the source already contains g

µν

(x); furthermore,

exact solutions are in general not known, and the approximation error could be such that the
corresponding error on S is larger than ¯

h. In this case we could still imagine that the zero-mode

can be computed in principle, but we would not have in practice any definite idea of its properties.
An explicit evaluation is therefore needed. To this end we consider static sources, with some free
parameters (typically m

+

, m

−

and their sizes), in the weak field approximation. We have in this

static case

S

zero

−mode

=

−

1
2

Z

d

4

x

q

g(x)g

µν

(x)T

00

(x)

(4)

Using the Feynman propagator one finds to first order in G (compare [3])

h

µν

(x) = 2G(2η

µ0

η

ν0

− η

µν

η

00

)

Z

d

3

y

T

00

(y)

|x − y|

(5)

2

It is straightforward to check from this expression that

p

g(x)g

00

(x) = 1 + o(G

2

). This means

that the action of the metric generated by a static source is simply

S =

−

1
2

Z

d

4

xT

00

(x) + o(G

2

)

(6)

and provided the integral of the mass-energy density vanishes, the field action is of order G

2

, i.e.

practically negligible, as shown in the following numerical example.

Consider a static dipolar source which is adiabatically switched on/off with a lifetime τ of

the order of 1 s. Suppose that the spatial size r of the source is of the order of 1 cm and the
two masses m

±

are of the order of 10

k

g, i.e. 10

37+k

cm

−1

in natural units. Note that with this

mass we have, for the ratio between the Schwarzschild radius and r, r

Schw.

/r

∼ 10

−29+k

. This

implies that we can compute h

µν

(x) in the weak field approximation, with negligible error. More

precisely, the residual of second order in G is found to be

S

order G

2

zero

−mode

∼ τ

G

2

m

2

±

r

3

∼ 10

−20+3k

(7)

Therefore the field of a static virtual source of this magnitude order, satisfying the condition

R

d

3

xT

00

(x) = 0, has negligible action even if k = 6, corresponding to an apparent matter

fluctuation with density 10

6

g/cm

3

. This should be compared to the action of the corresponding

“monopolar fluctuation”, namely S

monopolar

= (1/2)τ m + o(G

2

)

∼ 10

47+k

.

3

A Λ-term in the action cuts-off the fluctuations

The most recent estimates of the Hubble constant support a non-zero value of the cosmological
constant of the order of Λ

∼ 10

−50

cm

−2

. The cosmological term in the gravitational action, to

be added to the pure Einstein term, is

S

Λ

=

Λ

8πG

Z

d

4

x

q

g(x)

(8)

It is possible to evaluate the contribution of the dipolar fluctuations to this term. This is easier
for fluctuations with spherical symmetry, like those generated by virtual sources having the shape
of “+/- shells” (compare [3]). In this case one can use the exact Schwarzschild metric outside the
source and the spherically symmetric Newtonian field inside it. To leading order one finds that

∆S

Λ

=

Λτ

8πG

1
2

Z

source

d

3

xTrh(x) =

=

Λτ

4πG

Z

source

d

3

xV

N ewt.

(x) = Λτ mr

2

Q

(9)

where Q is an adimensional factor which can be negative or positive, depending on the distribution
of the positive and negative mass inside the virtual source.

Inserting for τ , m and r the same values as before, we find ∆S

Λ

∼ 10

−3+k

. Remembering

that the lower bound for pure gravity was k

∼ 6, we see that the cosmological term cuts, to some

extent, the dipolar fluctuations. This works even better for larger values of r.

3

4

Matter coupling vs. local changes in Λ

Let us consider now a scalar field φ coupled to gravity

L =

1
2

∂

α

φ∂

α

φ

− m

2

φ

2

(10)

T

µν

= Π

µ

φ∂

ν

φ

− g

µν

L = ∂

µ

∂

ν

φ

− g

µν

L

(11)

S

interaction

=

1
2

Z

d

4

x

q

g(x)T

µν

(x)h

µν

(x)

(12)

To lowest order in h

µν

the interaction action can be rewritten as

S

interaction

=

1
2

Z

d

4

x (h

µν

∂

µ

φ∂

ν

φ

− TrhL)

(13)

On the other hand, the cosmological action is, still to lowest order in h

µν

and expanding

√

g = 1 +

1
2

Trh + ...

S

Λ

=

Λ

8πG

Z

d

4

x

1 +

1
2

Trh

(14)

Therefore the sum of the two terms can be rewritten as

S

interaction

+ S

Λ

=

1
2

Z

d

4

xh

µν

∂

µ

φ∂

ν

φ +

1
2

Z

d

4

xTrh

Λ

8πG

− L

(15)

We see that to leading order the coupling of gravity to φ gives a typical source term

(h

µν

∂

µ

φ∂

ν

φ) and subtracts from Λ the local density 8πGL(x). This separation is arbitrary, but

useful and reasonable if the lagrangian density is such to affect locally the “natural” cosmological
term and change the spectrum of gravitational vacuum fluctuations corresponding to virtual mass
densities much larger than the real density of φ.

Let us give an example. Suppose that φ represents a coherent fluid with the density of

ordinary matter (

∼ 1 g/cm

3

). At the scale of 1 cm, with the observed value of Λ, the lower

bound on virtual source density is

∼ 10

3

g/cm

3

, which is much larger than the real density. If L

is comparable to Λ/8πG in some region, an inhomogeneity in the cut-off mechanism of the dipolar
fluctuations will follow, and this effect could dominate the effects of the coupling (h

µν

∂

µ

φ∂

ν

φ) to

real matter.

In our opinion, this dynamical mechanism could be the basis for an explanation of the

weak gravitational modification by superconducting spinning disks which E. Podkletnov claimed
to have observed under very special conditions [7, 8, 9]. A NASA/Argonne experimental team is
at work to replicate the original results, which at this time are neither confirmed nor confuted.

From the very beginning of our theoretical analysis [10] we maintained that the disks

described in Ref. [7] cannot be a source of gravitational field or perturbate it in a classical sense.
This is because the gravitational coupling to matter is far too small, and neither the presence of
Cooper pairs inside the disk nor its fast rotation help much under this respect. We have then
been looking for an interaction process not constrained by the coupling, and a candidate for this
are large quantum fluctuations (compare our phenomenological model in [11]).

More recently, anomalous gravity changes have been observed during a solar eclipse [12]. In

this case the coherent matter which couples to the gravitational fluctuations could be the native
iron which is abundant on the Moon.

4

5

Some remarks and new topics

(1) The fluctuations maintain

hT

µν

(x)

i = 0.

Even in the presence of strong quantum fluctuations, we expect that the vacuum average

of T

µν

(x), computed through the functional integral, is zero. This is defined as the average of

1

8πG

(R

µν

−

1
2

g

µν

R). More physically, there must be no apparent mass-energy- momentum density,

on the average, in the vacuum.

An analogous property holds in QED. Even in the presence of vacuum fluctuations, the

vacuum average value of the four-current, defined through Maxwell’s equations as j

µ

= ∂

ν

F

µν

,

vanishes.

In fact, some general properties of the functional integral ensure that the condition above

on T

µν

is respected, without any need of restricting the integration space [13]. In other words,

the fluctuations always average out in such a way to give a zero total virtual mass density at any
point.

(2) The dipolar zero-modes have S = 0 but are not extrema of S.

The dipolar zero-modes are not a minimum of the pure Einstein action, because this would

be equivalent of being a solution of the vacuum Einstein equations; but the dipolar zero-modes
do not satisfy the Einstein equations in vacuum, even through they have the same action as the
vacuum solutions (S = 0).

We can understand better the situation at hand through a bidimensional analogue (while,

of course, the space of the possible metric configurations is infinite-dimensional). Let us consider
the function f (x, y) = x

2

y

2

and draw its cartesian plot. We find that it resembles a paraboloid,

except for the fact that the axes x = 0 and y = 0 are a sort of “cuts”; along these axes the
function takes the value f = 0, which is also the value at the origin. However, the origin is a
minimum, while the points on the axes distinct from the origin are not minima.

(3) How much is the phase space volume of dipolar fluctuations?

The example above suggests another possible property of the dipolar zero-modes: in the

same way as the “zero lines” of the function above have null measure with respect to its full
bidimensional domain, one may think that the condition for the dipolar zero-modes defines a
subspace of all field configurations having lower dimension and thus null measure. In this case
the dipolar fluctuations would be suppressed because they have zero volume in phase space.

This would be true if the dipole condition m

+

= m

−

had to be satisfied exactly. In fact,

however, we considered weaker conditions. We saw, for instance, that up to values of the virtual
mass of the order of 10

6

g the difference between m

+

and m

−

, i.e., the width of the “line in phase

space”, is of order τ

−1

, where τ is the duration of the fluctuation.

The appearance of τ in this relation could also provide an upper bound on the duration of

the fluctuations, making τ closer to the minimum duration allowed by the Heisenberg principle,
like it happens for electromagnetic or scalar fields (compare [3, 4]). In other words, a long lifetime
τ requires a very precise compensation between the positive and negative masses of the virtual
dipole, thus implying a very small phase space volume for that configuration.

(4) Strong dipolar fluctuations are not allowed in QED.

This is an immediate consequence of the quadratic form of the electromagnetic lagrangian

density: L

∝ (E

2

− B

2

). The contribution to the action of the +/- charges in a virtual dipole

is the same, therefore there are no cancellations. The dipolar fluctuations are not favoured with
respect to the monopolar fluctuations.

5

The reasoning above also leads to the following statement.

(5) The null-action property of the dipolar fields is a non-perturbative feature.

In fact, suppose we limit ourselves to consider, as usual in perturbation theory, the part

of the action quadratic in h in an expansion around the minimum. We then would be in the
same situation as for the electromagnetic field. It is easy to see that the quadratic part of
the gravitational action is not positive definite and has positive and negative eigenvalues in the
Euclidean formulation, unlike the electromagnetic action [1]. Nevertheless, any single term would
be the same for positive or negative virtual sources (see eq. (5)), without any cancellation.

In other words, the null action property of the dipolar zero modes cannot be obtained just

considering the expansion in powers of h. This not because we are in strong field conditions, but
because the zero-modes are not local minima of the action (compare Point 2).

(6) The T

µν

of any real pure electromagnetic field generates gravitational zero-modes, up to terms

of order G

2

.

Since the trace of the energy-momentum tensor of any electromagnetic field is zero, it is

possible to use, instead of virtual unphysical dipolar mass sources, a real electromagnetic field
as source of gravitational zero- modes. It needs to be a pure electromagnetic field in vacuum,
thus any plane waves or wave packets are admissible. We stress again, however, that the real or
virtual character of the source is irrelevant in order to obtain vacuum fluctuations.

Acknowledgments - This work was supported in part by the California Institute for Physics
and Astrophysics via grant CIPA-MG7099. The author is grateful to C. Van Den Broeck, M.
Gross and J. Brandenburg for useful discussions and remarks.

References

[1] Modanese, G. (1998) Stability issues in Euclidean quantum gravity, Phys. Rev. D 59, 024004.

[2] Modanese, G. (1999) Virtual dipoles and large fluctuations in quantum gravity, Phys. Lett.

B 460, 276-280.

[3] Modanese, G. (2000) Large “dipolar” vacuum fluctuations in quantum gravity, Nucl. Phys.

B 588, 419.

[4] Modanese, G. (2000) Paradox of virtual dipoles in the Einstein action, Phys. Rev. D 62,

087502.

[5] Wheeler, J.A. (1957), Ann. Phys. 2, 604.

[6] Hamber, H.W. (2000) On the gravitational scaling dimensions, Phys. Rev. D 61, 124008.

[7] Podkletnov, E. and Nieminen, R. (1992) A possibility of gravitational force shielding by

bulk Y Ba

2

Cu

3

O

7

−x

superconductor, Physica C 203, 441; Podkletnov, E. (1997) Weak

gravitational shielding properties of composite bulk Y Ba

2

Cu

3

O

7

−x

superconductor below

70 K under e.m. field, report cond-mat/9701074.

[8] Modanese, G. and Schnurer, J. (1996) Possible quantum gravity effects in a charged Bose

condensate under variable e.m. field, report gr-qc/9612022.

[9] For a recent theoretical paper with several references see Ummarino, G.A. (2000) Possible

alterations of the gravitational field in a superconductor, report cond-mat/0010399.

6

[10] Modanese, G. (1996) Theoretical analysis of a reported weak gravitational shielding effect,

Europhys. Lett. 35, 413-418.

[11] Modanese, G. (1999) Gravitational anomalies by HTC superconductors: a 1999 theoretical

status report, report physics/9901011.

[12] Qian-shen Wang et al. (2000) Precise measurement of gravity variations during a total solar

eclipse, Phys. Rev. D 62, 041101.

[13] Collins, J.C. (1984) “Renormalization” (Cambridge University Press, Cambridge, 1984).

Modanese, G. (1994) Vacuum correlations at geodesic distance in quantum gravity, Riv.
Nuovo Cim. 17, Vol. 8, 1-62.

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